Publications of (and about) Paul Erdös
Autor: Erdös, Paul; Hajnal, András; Milner, E.C.
Title: Set mappings and polarized partition relations (In English)
Source: Combinat. Theory Appl., Colloquia Math. Soc. Janos Bolyai 4, 327-363 (1970).
Review: [For the entire collection see Zbl 205.00201.]
A set mapping on a set S is a function f from S into the set of subsets of S such that x \not in f(x) (x in S); A \subset S is called a free set (for the set mapping) if y \not in f(x) for all x,y in A, i.e. A \cap f(A) = Ø.
In this paper we shall consider set mappings on a well-ordered set S in the case when the order type of S is not necessarily an initial ordinal. In particular, we examine the truth status of the following statement SM(\alpha , \lambda). If f is any set mapping of order \alpha on a set type \lambda, then there is a free subset having the same order type \lambda. The Erdös-Specker generalization of the Ruziewicz conjecture asserts that SM(\alpha,\lambda) holds if \lambda is an infinite initial ordinal and \alpha < \lambda. We only examine the problem for the case when |\lambda| = \aleph1 although some of our results hold more generally. We will prove that SM(\alpha,\lambda) holds in the following cases:
(i) \alpha < \omega1 and \lambda = \omega\sigma1+11+...+\omega\sigmak+11 < \omega\omega+21 (k finite);
(ii) \alpha = \omega0 and \lambda = \omega1\gamma < \omega\omega+21;
(iii) \alpha < \omega0; \lambda = \omega\Theta, where \Theta is arbitrary.
Note that the form given for \lambda in (i) is the most general for which SM(\alpha,\lambda) is true with any \alpha < \omega1.
Classif.: * 04A20 Combinatorial set theory
Citations: Zbl 205.00201(EA)
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